\chapter{Lipids and membranes}
%\chapter{Lipids, membranes, and cellular transport}
\label{ch:biophys}
%\begin{flushright}\emph{}\\- \end{flushright}


Lipids\footnote{From the Greek {\it lipos}, meaning ``fat''.} are the fundamental building blocks of any biological membrane~(Figure~\ref{fig:memECM}); they are generally defined as substances that are soluble in organic solvent but only sparingly soluble in water.~\cite{voet} % 383
\begin{figure}%[ht]
\centering
\includegraphics[scale=.5]{biophys/memECM.eps}
\caption[The plasma membrane]{The plasma membrane. Schematic representation of a plasma membrane patch at molecular resolution; characteristic elements are highlighted.~\cite{simmons}}
\label{fig:memECM}
\end{figure}
As a class of molecules, lipids display a wide diversity in both structure and biological function, and they self-organise into many intriguing structures, with extraordinary material properties that have been optimised by evolutionary principles over billions of years.~\cite{dowhan,mouritsen} 
Lipid membranes play crucial roles in compartmentalisation, they represent a solvent for other species (such as proteins) to function, and they contribute to the structural scaffolding of cells. Along with these rather passive functions, it has recently become evident that lipids can actively contribute to fission and fusion events, and in general to the regulation of membrane proteins. It is now accepted that lipids are as important for life as proteins, sugars, and genes.~\cite{mouritsen}
In this chapter, the fundamental properties and phenomena regarding lipids and lipid bilayers are summarised, along with some of the most popular experimental techniques employed to study membrane systems. %Particular emphasis is placed on the phospholipid bilayer and transport therein. %, which are we propose the model described in the subsequent chapters of this thesis.   

\section[Types of membrane lipids]{Types of membrane lipids}

 There are three common classes of membrane lipids: phospholipids, glycolipids and cholesterol. Phospholipids are the major class of membrane lipids,~\cite{berg} %p322
and are the main object of the modelling work presented in this thesis. Hence, in the following sections, we will focus on this particular lipid species. 
However, we give here a brief account on the two other common lipid types. 

{\em Glycolipids}, as their name implies, are sugar-containing lipids. They are formed by association of a carbohydrate chain with lipids on the external surface of the cell membrane. Glycolipids thus extend from the plasma membrane into the aqueous environment outside the cell, acting as a recognition site for specific chemicals as well as helping to maintain the stability of the membrane and attaching cells to one another to form tissues. They also serve a role as energy stores.

{\em Cholesterol} is a lipid with a unique structure compared with the other lipids; it is a steroid, built from four linked hydrocarbon rings, terminated at one end by a hydrocarbon tail and on the other by a hydroxyl group. Cholesterol is universally present in the plasma membranes of all animals, in ratios of 25-50$\%$ of the total lipid content; however, it is essentially absent from some intracellular membranes, such as mitochondrial and Golgi.~\cite{mouritsen} %p325
Cholesterol is important in stabilising membranes, as its presence makes them thicker and less leaky; it is also an essential component of lipid ``rafts'', membrane microdomains believed to favour specific protein-protein interactions resulting in the activation of signalling cascades.~\cite{simons00}



\section[Phospholipid structure and self-assembly]{Phospholipids: structure and self-assembly}
 The most important class of lipids in natural membranes is represented by the {\em phospholipids}.~\cite{berg} Phospholipids are amphiphilic %\footnote{From the Greek {\it amphi}, that means ``on both ends'', and {\it philos}, that means ``loving''.} 
molecules constructed from four components: fatty acids, a ``backbone'' to which the fatty acids are attached, a phosphate, and an alcohol attached to the phosphate. The fatty acid components constitute a hydrophobic barrier, whereas the rest of the molecule has hydrophilic properties to enable interaction with the environment. The backbone component may be {\em glycerol}, a 3-carbon alcohol, in which case the phospholipid is called {\em phosphoglyceride}, or it may be {\em sphingosine}, a (more complex) amino alcohol, which constitutes {\em sphingolipids}. Sphingolipids comprise important lipid species such as {\em ceramide}, a fundamental constituent of the skin and an important molecule involved in programmed cell death.~\cite{mouritsen} 
 However, the main lipid components of biological membranes are {\em phosphoglycerides} (or {\em glycerophospholipids}).~\cite{voet} %p.385
Figure~\ref{fig:pl} 
\begin{figure}%[ht]
\centering
\includegraphics[scale=.5]{biophys/phospholipid.eps}
\caption[Phospholipid molecule]{Phospholipid molecule. It is possible to identify the polar head group, the glycerol moiety, the ester groups and the hydrocarbon chains.\cite{ualr}}%/~botany.}
\label{fig:pl}
\end{figure}
shows a typical phosphoglyceride, in particular a {\em phosphatidylcholine} lipid: the polar head comprises a positive group (choline (CH$_3$)$_3$N-CH$_2$-CH$_2$) and a negative group (phosphate O-PO$_2$-O), whereas the hydrophobic part is made of the glycerol backbone (CH$_2$-CH-CH$_2$), two ester groups (O-CO-CH$_2$) and two hydrocarbon ``chains'' (also called ``tails'') of fatty acids. Hydrocarbon chains are formed by consecutive methylene (CH$_2$) segments terminating with a methyl (CH$_3$) segment. Hydrocarbon tails often comprise one or more double bonds between C atoms along the tail. 
In general, when dispersed in water, lipids are driven together by the {\em hydrophobic effect}, and self-organise in various aggregates depending on the specific lipid structure, temperature and level of hydration (Figure~\ref{fig:lipStructs}~a).
\begin{figure}
\centering
\mbox{\subfigure[]{\includegraphics[width=70mm]{biophys/lipidStructures.eps}}\quad
      \subfigure[]{
	\includegraphics[width=40mm]{biophys/vesicle.eps}
}}
\caption[Lipid aggregates]{The most common  aggregates formed by lipid molecules. (a) Non-bilayer phases.~\cite{scq} (b) Lipid bilayer vesicle. Water molecules, hydrating the interior and exterior of the vesicle, are also schematically represented.~\cite{steve}}
\label{fig:lipStructs}
\end{figure}
 The hydrophobic effect, that is, the tendency for oil and water to separate, is a complex phenomenon that manifests different characteristics depending on the system's temperature and pressure, the shape of the oil-like components and the size of the aggregates involved.~\cite{southall02,chandler05} In the particular case of lipid/water mixtures at biological temperature and pressure, the hydrophobic effect is believed to be mainly of entropic origin.~\cite{mouritsen}  Pure water systems are characterised by a tetrahedral arrangement of hydrogen-bonded molecules which maximises the system's entropy.  When insoluble species (such as hydrocarbon molecules) are introduced in water, a loss in entropy is produced due to the induced local ordering of water around every insoluble molecule. To reduce such ordering, and hence to minimise entropy loss, the insoluble molecules are driven together so that the total surface area exposed to water, corresponding to the total area involved in local ordering phenomena, is decreased. This gain in entropy of the water upon assembly formation outweighs the enthalpy penalty (caused by the demixing of water and the insoluble species) and the loss of configurational entropy of the insoluble molecules due to the constraints typically imposed by the aggregate structure.\cite{hamley}
Owing to the fact that lipids form assemblies  by self-aggregation processes that do not involve strong chemical forces, such assemblies can be categorised as {\em soft matter} materials. Soft matter comprises a large and ubiquitous class of systems including for example polymers, emulsions, colloids, liquid crystals and many biological materials. All these systems exist in a condensed phase which cannot be described unambiguously as either liquid or solid, as it typically possesses mixed properties. For instance, soft matter may display long-range ordering properties typical of solid matter. However, unlike conventional solid materials, soft materials have physical properties which are largely dominated by entropy. Soft matter is highly deformable, and typically constructed in a hierarchical manner with substructures subtly interacting on several length and time scales.~\cite{mouritsen} All aggregates formed by lipids represent  specific examples of soft matter structures.
From a biological perspective,  {\em bilayer} structures  (Figure~\ref{fig:lipStructs}~b) are considered the most important category of  lipid assemblies,  as they form the fundamental backbone of the majority of biological membranes.
%Many cellular properties are consequences of the pfundamental physical principles of self-organisation that rule when many molecules act in concert. A key player in this concert is water, which functions as the unique biological solvent: the peculiar properties of water force lipid molecules to self-assemble and organise into subtle structures, as for instance bilayer membranes. 
%: softness is a feature that lipid membranes share with other forms of condensed matter, like polymers and liquid crystals.~\cite{mouritsen} 
%The hydrophobic effect arises from the disruption of the water hydrogen bonding network caused by the presence of hydrophobic molecules: these destibilise the hydrogen bonding network thus lowering the entropy of the system. The hydrophobic effect acts to drive the hydrohpbic molecules together to minimise the contact with water and hence to restore as much as possible the hydrogen bonding network.~\cite{mouritsen} 
%From a vesicle, one can imagine to isolate a small bilayer patch 
 %The molecules that play the dominant roles in membrane formation all have higly polar head groups and, in most cases, {\em two} hydrocarbon tails\footnote{There is a molecular sense to this. If a large headgroup is attached to a single hydrocarbon chain, the molecule is wedge-shaped and will tend to form spherical micelles. A double tail yields a roughly cylindrical molecule, which can easily pack in parallel to form extended sheets of bilayer membranes~\cite{mathews}.}.
%Indeed geometric factors dictate that for most two-chain  phospholipids, bilayers are the favoured structure, rather than micelles or inverted hexagonal phases~\cite{cevc}. % p.2
%The occurrence of phospholipids as an essential membrane component is attributable to their ability to form bilayer vescicles spontaneously when dispersed in water~\cite{cevc}. 
%In the remainder of this chapter, we will focus on the lipid bilayer, which forms
\section{Phospholipid bilayers}

Phospholipid bilayers constitute the basic material employed to encapsulate the cell and its sub-compartments. 
In the following sections, the main features of lipid bilayers are summarised.

\subsection{Structure}

%The phospholipid bilayer is the structural foundation of biomembranes and 
Structural data on lipid bilayers are widely used as basic information to help understand and model biomembrane structure and the functions that take place therein.~\cite{nagle00a} % Reliable experimental data, though incomplete, also provide a guide to modelling and a necessary check on the reliability of simulations.~\cite{nagle00a} 
%Measurements of the volume per lipid can be performed using different techniques. The simplest method employs neutral flotation: the density of the aqueous solvent is varied by mixing D$_2$O with H$_2$O, with the density of the lipid being given by the density of the aqueous mixture in which the bilayers neither sink nor float.~\cite{nagle00a} % p.163
%The internal structure of bilayers is studied by neutron or X-ray diffraction methods. 
Most %diffraction 
experimental studies are performed on stacks of hydrated bilayers, especially on multi-lamellar vesicles (Figure~\ref{fig:mlvs}).
\begin{figure}%[ht]
\centering
\includegraphics[scale=.7]{biophys/mlvs.eps}
\caption[Multi-lamellar vesicles]{Schematic representation of multi-lamellar vesicles. The black areas represent regions of excess water. From Koenig et al.~\cite{koeni97a}}
\label{fig:mlvs}
\end{figure}
The internal structure is investigated by X-ray, neutron scattering, molecular-probe, and magnetic resonance techniques.
The measurements obtained provide information about the membrane thickness, and, most importantly, can resolve the depth-dependent distribution of specific lipid segments across the bilayer (Figure~\ref{fig:bilayerEdp}).
\begin{figure}
\centering
\includegraphics[scale=1.1]{biophys/bilTransStruct.eps}
\caption[Transbilayer structure]{Transbilayer structure. The top panel shows a phospholipid bilayer model; hydrogens are coloured in white, oxygens in red, carbons in turquoise, nitrogens in blue and phosphoruses in dark yellow (adapted from Feller~\cite{feller-dppc}).  The bottom panel shows  corresponding density profiles obtained from neutron-scattering and X-ray techniques; the curves give the relative probabilities of finding the different molecular segments of the phospholipid molecules (adapted from Nagle and Tristram-Nagle~\cite{nagle00a}).}
\label{fig:bilayerEdp}
\end{figure}

\subsubsection{Structure of the hydrocarbon region: intramolecular order parameters}
The hydrocarbon tail region of a bilayer can be investigated by deuterium magnetic resonance; this technique allows tail ordering to be quantified in terms of order parameters.  Order parameters generally describe the tail orientation as a function of depth. %;~\cite{jseelig74} they also provide a basis for understanding the concept of membrane fluidity.~\cite{douliez95}
%The middle panel of~Figure~\ref{fig:stevBeadSpring} offers a schematic descritpion: the molecular axis $Z_D$ is normal to the plane defined by the CH bonds.
%Experiments are carried out after deuteration, so the actual bond vectors can also be called CD. 
For each methylene group $k$ along a lipid tail, the intramolecular order parameter $S^k_\textrm{CD}$ can be defined as:~\cite{akuts91a}
\begin{equation}
S^k_\textrm{CD} = \langle 3\cos^2\theta-1 \rangle / 2
\end{equation}
with $\theta$ the instantaneous angle between the $k$-th C$-^2$H bond vector and the overall molecular axis. The overall molecular axis is the main axis of a lipid molecule, which, on average in the biologically-relevant fluid phase, is parallel to the ``bilayer normal'', that is, the direction perpendicular to the bilayer plane.~\cite{akuts91a}
 It is also possible to define the molecular axis of a chain segment $k$ %(either a CH$_2$ or a CH$_3$ group) 
as the normal direction to the plane spanned by the two CH bonds of the $k$-th methylene group.~\cite{jseelig74} For each methyl segment $k$, the intramolecular {\em segmental} order parameter $S^k_\textrm{mol}$ is then:
\begin{equation}
S^k_\textrm{mol} = \langle 3\cos^2\eta -1\rangle / 2
\end{equation}
with $\eta$  the instantaneous angle between the 
 molecular axis of the $k$-th segment and the bilayer normal.  $S^k_\textrm{mol}$ are thus the order parameters of the segments' molecular axes with respect to the %director; they represent the fluctuation of the molecular axis around the 
bilayer normal. %~\cite{akuts91a} 
 Experiments can accurately determined $S^k_\textrm{CD}$, which can then be related to the segmental order parameters $S^k_\textrm{mol}$ using the formulae:~\cite{jseelig74}
\begin{displaymath}
  S^k_\textrm{mol} = 
\left\{ \begin{array}{cl} -2\,S^k_\textrm{CD}  & \textrm{for the {\em k}-th CH$_2$ segment}\\ 
-3\,S^k_\textrm{CD}  & \textrm{for the terminal CH$_3$ segment %(where for example $k=14$ for DMPC)
}
\end{array} \right. 
\end{displaymath} 
In general, $S^k_\textrm{mol} = 0$ indicates a completely random mean orientation, $S^k_\textrm{mol} = 1$ indicates alignment of the segment molecular axis along the bilayer normal, whereas $S^k_\textrm{mol} = -0.5$ indicates that the segment molecular axis lies in the bilayer plain (thus being perpendicular to the normal direction).
 
\subsection{Phase behaviour of lipid bilayers}
Lipids are able to adopt a range of phases depending on temperature and level of hydration, as already mentioned. In particular, fully hydrated bilayers composed of a single phospholipid species undergo a well-defined thermotropic phase transition in which the lipid chains change from an ordered, or gel, state to a fluid, or liquid-crystalline, state.~\cite{cevc} % p.12
As an example, the phase diagram for the dimyristoylphosphatidylcholine (DMPC) bilayer is reported in Figure~\ref{fig:dmpcPhaseDiag}.
\begin{figure} 
\begin{center}
\includegraphics[scale=.2]{biophys/dmpcPhaseDiagram_Janiak79}\caption[DMPC phase diagram]{Phase diagram of hydrated DMPC bilayers, together with representations of the L$_\alpha$, P$_{\beta'}$ and  L$_{\beta'}$ phases. The hydrocarbon chain packing is a hexagonal array for the  P$_{\beta'}$ phase and a ``distorted'' hexagonal lattice for the  L$_{\beta'}$ phase. From Janiak et al.~\cite{janiak79}}
\label{fig:dmpcPhaseDiag}
\end{center}
\end{figure}
Biologically, the most important phase is the liquid $L_\alpha$ phase, characterised by a high degree of disorder in the alkyl chains of the hydrophobic core. 


%\subsection{Mechanical properties}
\subsection{The lateral pressure profile}
%Lipid bilayers are generally tensionless: the repulsive (inter-tail and inter-headgroup) interactions exactly balance the attractive (``hydrophobic'') forces.~\cite{harries97}
The transbilayer lateral pressure profile $\pi(z)$, where $z$ is a spatial coordinate along the bilayer normal, is defined as the difference between the lateral and the normal pressures acting inside the bilayer: $\pi(z) = p_L(z) - p_N(z)$. The lateral pressure profile thus characterises the transmembrane distribution of forces; Figure~\ref{fig:lppTempler} is an example of the proposed shape of such a distribution.
\begin{figure} 
\begin{center}
\includegraphics[scale=.9]{biophys/lppTempler}
\caption[Lateral pressure profile]{Lateral pressure profile. Proposed distribution of lateral pressure $\pi$ within a flat monolayer as a function of the position along the interfacial normal $z$. From Templer et al.~\cite{templer98}}
\label{fig:lppTempler}
\end{center}
\end{figure}
In terms of magnitude, peak pressures of the order of several hundreds of atmospheres are predicted; the pressure profile results from an interplay of enormous opposing forces,  that ultimately compensate each other.
The lateral pressure profile changes in relation to the lipid composition (or the state of the lipid headgroups, e.g., by proton or ion binding), and as a result of the presence of cholesterol or solutes (such as drugs); the consequent depth-dependent changes in the stress distribution are predicted to affect lipid phase behaviour and the conformation of inclusions such as proteins.~\cite{sedd95} %[Sec.~6.4].
In fact, the lateral pressure profile controls a very large number of membrane features and phenomena: it determines the interfacial area, it is at the basis of phase transitions and fusion,~\cite{sedd95,yang03,kozlovsky04,siegel04,shearman06} %seddon @p.144 %
it affects  permeability,~\cite{kamo06}  drug transport,~\cite{curnow04} and anaesthesia,~\cite{mohr05} 
it modulates the insertion and folding of membrane proteins,~\cite{curran99,meijberg02,brink04,hong04,bowie05}
 and it is believed to directly control the functioning of several membrane proteins, such as the lipid synthesis regulatory enzyme CCT,~\cite{attard00,davies01} diacylglycerol kinase,~\cite{fanani04} phospholipase A2~\cite{sen91} and C,~\cite{ruiz98} rhodopsin,~\cite{botelho02,wang02,botelho06} and several transbilayer channels.~\cite{keller93,perozo02,jensen04,brink04,rosto06}
%~\cite{mouritsen},  channel conductance~\cite{keller93}, mechanosensitive channels~\cite{perozo02} and the potassium channel KcsA~\cite{brink04}.   
%Despite so many effotrs, there are many apparently simple questions about lipids that still need to be answered~\cite{bagatolli06}: for instance, why do cell membranes contain thousands of different molecular lipid species?
The lateral pressure profile is also directly related to the elastic curvature constants that characterise the Helfrich expression for the bending free energy.~\cite{helfrich73,sedd95,marsh06} %~\cite{helfrich73}. 
According to Helfrich's theory, the surface curvature elastic energy per unit area $g$ is concisely expressed as: \begin{equation}\label{eq:helfrich}
g = \kappa\left(c_1 + c_2 -c_0\right)^2/2 + \kappa_\textrm{G}\, c_1\,c_2 
\end{equation}
with $\kappa$ the bending rigidity, $c_1$ and $c_2$ the (local) principal curvatures,  $c_0$ the spontaneous (or intrinsic) curvature and $\kappa_\textrm{G}$ the Gaussian curvature modulus.  Equivalently, the Helfrich equation can also be written:
\begin{equation}\label{eq:helfrichBis}
g = 2\kappa\left(H-H_0\right)^2+ \kappa_\textrm{G}K 
\end{equation}
with $H=(c_1+c_2)/2$ the mean curvature, $H_0 = c_0/2$ the equilibrium mean curvature and $K=c_1c_2$ the Gaussian curvature. The constants appearing in Helfrich's expressions in turn control membrane shape, %~\cite{zim-koz06}, 
 and play specific roles  in the mechanisms modulated by the lateral pressure profile (mathematical relations between the Helfrich constants and the pressure distribution $\pi(z)$ can be found elsewhere.~\cite{mario08,marioDmpcDopc}  

Experimentally, it has been so far impossible to quantitatively measure the pressure profile. The only experiments performed to date have yielded qualitative and partial pictures for the hydrocarbon region only. In these experiments,~\cite{templer98, kamo06} changes in the lateral pressure along the bilayer normal were ``sensed'' using a series of di-pyrenyl phosphatidylcholine (dipyPC) fluorescence probes. DipyPCs are PC lipids carrying pyrene moieties attached to their tail ends. Ultraviolet stimulation produces both monomer and excimer fluorescence from pyrene. The excimer signal, which is entirely intramolecular at low dilutions of dipyPC,  results from (excited) dimerization of adjacent pyrene groups, and depends on the frequency with which the two pyrene moieties collide to form excimers; this frequency, in turn, is proportional to the lateral pressure. % are brought into close proximity (aggregation frequency). 
The relative intensity of the excimer to monomer signal is thus a measure of the pressure; by using dipyPCs of different acyl chain lengths it is possible to qualitatively estimate the pressure variations across different depths in the bilayer.~\cite{templer98}  

\subsection{The dipole potential}
In typical physiological conditions, the presence of ions in the water phases at the interface with both sides of a membrane, along with the orientational ordering of interfacial water dipoles and the intramembrane distribution of charged groups along the lipids (Figure~\ref{fig:chgsDipoles}), create a characteristic electrical potential distribution along the direction normal to the membrane plane (Figure~\ref{fig:proposed-epp}).
\begin{figure} 
\begin{center}
\includegraphics[scale=.15]{biophys/lipWatElectrostatics}\caption[Charges and dipoles of lipids and water]{Electrostatics at the lipid/water interface. The signs and arrows represent the main charges and dipoles possessed by lipid segments and water molecules. Adapted from Shinoda et al.~\cite{shinoda98}}
\label{fig:chgsDipoles}
\end{center}
\end{figure}
%\begin{figure}  \begin{center} \includegraphics[scale=.3]{biophys/electricalPotential_Clarke.eps}\caption[Trans-bilayer electrical potential]{The electrical potential across a phospholipid membrane. The trans-membrane potential $\Delta\Psi$ is due to the difference in anion and cation concentrations between the two aqueous bulk phases. The surface potential $\Psi_s$ arises from charged residues (charged headgroups) at the membrane-solution interface. The dipole potential $\Psi_d$ results from the alignment of dipolar residues of the lipids and associated water molecules within the membrane. From Clarke.~\cite{clarke01}} \label{fig:proposed-epp} \end{center} \end{figure} 
\begin{figure}  \begin{center} \includegraphics[scale=1.2]{biophys/elPot.eps}\caption[Trans-bilayer electrical potential]{The electrical potential profile $\Psi$ across a phospholipid membrane. The trans-membrane potential $\Delta\Psi$ is due to the difference in anion and cation concentrations between the two aqueous bulk phases. The surface potential $\Psi_s$ arises from charged residues (charged headgroups) at the membrane-solution interface. The dipole potential $\Psi_d$ results from the alignment of dipolar residues of the lipids and associated water molecules within the membrane. From Clarke.~\cite{clarke97}} \label{fig:proposed-epp} \end{center} \end{figure}
In particular, the membrane {\em dipole potential} $\Psi_d$ %is an electrical potential which 
originates from the alignment of dipolar residues of the lipids and water dipoles in the bilayer-water interfacial region.
 $\Psi_d$ is positive inside the membrane with respect to the outer water phase; its exact value is unknown, but it is believed to be of the order of $0.2-0.5$\,V.  Since this potential drops across a very small distance within the headgroup region of the membrane,~\cite{clarke01} the corresponding electric field strength is enormous, with peak values in the range $10^8-10^9$ V/m.  %the much higher field strength associated with the dipole potential would certainly be capable of affecting the orientation of dipolar or charged protein segments which are located in the interfacial region of the membrane; the only factor that could lessen this effect would be an electrical shielding from the protein by other charged residues.~\cite{clarke01} 
The membrane dipole potential, and associated electric field, are involved in a great number of biological processes, such as  membrane fusion,~\cite{cladera99,cladera01} permeation,~\cite{franklin93} the regulation of  membrane proteins (Na$^+$,K$^+$-ATPase,~\cite{starke05} gramicidin channel,~\cite{rokitskaya02} phospholipase A$_2$~\cite{maggio99}) insertion and folding of amphiphilic peptides,~\cite{cladera98} the kinetics of DNA-lipid complexes,~\cite{gelbart00} % from Saiz and Klein, jcp, 2002
the kinetics of redox reactions at membrane surfaces,~\cite{alakoskela01} human skin permeability,~\cite{cladera03} general anaesthesia,~\cite{qin95,cafiso98} membrane partitioning of pregnanolone,~\cite{alakoskela04} the binding capacity of saquinavir,~\cite{asawakarn01} and the modulation of molecule-membrane interactions in lipid rafts with possible effects on cells signalling.~\cite{luker01,oshea03}  
%anesthesia and %~\cite{cafiso98} and %the modulation of molecule-membrane interactions in lipid rafts with effects on and signalling.~\cite{starke06} % luker01
Despite the growing evidence for its importance, the dipole potential $\Psi_d$ has received so far relatively little attention;~\cite{clarke01} for instance, the overall transmembrane potential~$\Delta\Psi$, which regulates numerous ion channels, is much more popular.  The reason is that, while $\Delta\Psi$ can be easily measured and controlled by placing electrodes in the solution phases on each side of the membrane, the dipole potential $\Psi_d$ cannot be directly measured, as it is impossible to insert electrodes at different depths within the membrane.~\cite{clarke01} Therefore, the dipole potential can only be estimated by indirect measurements. One method involves studying the membrane conductivity associated with the translocation of different hydrophobic ions; due to the presence of the dipole potential, hydrophobic anions permeate much faster than hydrophobic cations, and the magnitude of this effect can be used to quantify~$\Psi_d$.~\cite{gawrisch92,schamberger02} It is also possible to consider the bilayer dipole potential to be equivalent to the potential across a monolayer, which can be directly obtained using electrodes after spreading a monolayer of lipids onto the surface of a Langmuir trough.~\cite{lairion04} However, this measurement relies on the questionable assumption that such isolated monolayers are equal to each of the monolayers paired into a bilayer assembly. Recently, the dipole potential has been estimated using cryo-EM, by recording the interactions of electrons with regions of different electrostatic potentials across rapidly frozen bilayers.~\cite{wang06} Unfortunately, this technique also relies on a number of approximations that might affect its reliability. For instance, it is expected that the bilayer structure remains intact during the freezing, which occurs at a rate of~$10^6\,$K/s $=1\,$K/$\mu$s; in fact, this cooling rate might be slow enough to allow artificial rearrangements of water and lipid molecules. The available experimental estimates for the dipole potential of ester-PC lipids are collected in Table~\ref{tab:dipPotExp}.
\begin{table}
\caption[Dipole potential: experimental measurements]{Measurements of the dipole potential $\Psi_d$ in phosphocholine bilayers.}
\begin{center}
%\vspace{8pt}
\begin{tabular}{|l|c|c|} % put @{} if you want to eliminate horizontal space between columns
\hline
{\em Method} & {\em Lipid} &  $\Psi_d$  / V \\\hline
Ion translocation~\cite{gawrisch92} & DPPC & 0.227\\
Ion translocation~\cite{schamberger02} & DPPC & 0.346\\
Monolayer~\cite{lairion04} & DMPC & 0.449 \\
Cryo-EM~\cite{wang06} & DPhPC & 0.510 \\\hline
%Atomic-level MD~\cite{feller96a} & DPPC & 0.800-2.300\\
%Atomic-level MD~\cite{berge97a} & DPPC & 0.150-0.750\\
%Atomic-level MD~\cite{smond99a} & DPPC & 0.600\\
%Atomic-level MD~\cite{mashl01} & DOPC & 0.500\\Atomic-level MD~\cite{saizl02b} & DMPC & 0.500\\
%Atomic-level MD~\cite{anezo03a} & DPPC & 0.620-0.830\\
%Atomic-level MD~\cite{patra03a} & DPPC & 0.570 \\
%Atomic-level MD~\cite{sachs04} & DMPC & 0.950\\
%Atomic-level MD~\cite{shinoda04} & DPhPC & 1.002\\
%Atomic-level MD~\cite{wohle04a} & DPPC & 0.500-0.850\\
%Atomic-level MD~\cite{villarreal04} & DPPC & 0.557\\
%Atomic-level MD~\cite{song05} & DPPC & 0.8\\
%Coarse-grain MD~[{\em this work}] & DMPC & 1.2\\
\end{tabular}
\label{tab:dipPotExp}
\end{center}
DPPC = dipalmitoylphosphatidylcholine, DMPC =  dimyristoylphosphatidylcholine, DPhPC = diphytanoylphosphatidylcholine.
\end{table}

\subsection{Dynamics}
Despite displaying structural integrity typical of solid materials, the biologically relevant state of lipid bilayers is that of a liquid crystal characterised by substantial fluidity and disorder. 
In their biological state, lipids typically display various dynamic features: they change conformation, diffuse laterally in the membrane plane, rotate around their main molecular axis, protrude into the water phase and flip through the monolayers. These characteristic motions are represented in Figure~\ref{fig:lipDyn}. 
\begin{figure}%[ht]
\centering
\includegraphics[scale=1]{biophys/lipMotion.eps}
\caption[Lipid characteristic motions]{Lipid dynamics. Characteristic motions of lipid molecules inside a bilayer: a)~tail conformational change, b)~rotation around molecular axis, c)~diffusion (swap), d)~protrusion, e)~flip-flop. From Mouritsen.~\cite{mouritsen}}
\label{fig:lipDyn}
\end{figure}
The lateral mobility of lipids in the plane of the membrane (diffusion) is particularly important, as it defines the  liquid-like nature of membranes.
The diffusion of lipids can be measured by a number of different experimental methods. The motion of a single lipid can be detected by single-particle tracking.~\cite{sonnleitner99} A colloidal particle of a typical diameter of 40\,nm is attached to the lipid molecule and the particle's motion is followed by microscopy; the spatial resolution of this kind of experiment is~$\approx50$\,nm, and the time resolution is~$\approx5$\,ms.  
Diffusion coefficients have been obtained by quasi-elastic neutron scattering, which measures  short-range diffusion taking place over sub-nanosecond timescales.~\cite{tabony91} Alternatively, long-range methods such as NMR spectroscopy~\cite{filip03b} and fluorescence recovery after photobleaching~\cite{almeida92} can be used; these methods probe millisecond time-scales. Interestingly, the lateral diffusion coefficients measured with short-range methods turn out to be~$\approx2$ orders of magnitude higher than those obtained by long-range observations.~\cite{vaz91} 
To understand the reason of this discrepancy, it is useful to take into account the {\em free-volume} diffusion theory, which was  proposed more than 50\,years ago to describe  transport in soft condensed matter.~\cite{cohen59, turnbull61, turnbull70} The foundation of the theory is that diffusion is related to the average free volume per particle; molecular diffusion proceeds by jumps occurring when a large enough free volume is created (by free volume redistribution) next to the diffusing molecule.
Lipid diffusion is then assumed to proceed by ``hopping'' of molecules into vacancies formed by lateral density fluctuations in the membrane.~\cite{oleary87}
  %,almeida92} 
%According to the free-volume model, lateral diffusion occurs by discrete jumps of lipid molecules %, of length roughly equal to the diameter of a lipid molecule, into nearby vacancies formed by lateral density fluctuations; 
In between jumps, a lipid molecule spends a relatively long time ``rattling in a cage'' formed by its neighbours. Over short ($<1$\,ns) time-scales, the diffusion coefficient is high because it is determined by the rapid short-range lipid motion mainly due to such ``rattling-about'' behaviour. % it is also partly due to lipid jumps, although over such short time-scale the motion of rattling vs jumping lipids cannot be discriminated. 
Over longer times, this rattling motion averages out, yielding no net displacement. The true, long-range diffusion coefficient is thus determined by the lipid jumps, that give rise to effective displacement over extended ($>10$\,ns) time-scales. 

\section{Summary}


Membrane lipids are driven together by water, through the hydrophobic effect, into various self-assembling structures. The most abundant lipid aggregate is the bilayer, which constitutes the backbone of the plasma membrane encapsulating cells. Owing to many peculiar  molecular characteristics of lipids (such as shape and charge anisotropy) and to the fact that they interact through ``weak'', non-covalent forces, the resulting bilayers are multifaceted materials:
\begin{itemize}
\item the internal structure is highly stratified and characteristically (dis)ordered;
\item there is a steeply varying depth-dependent distribution of pressures;    
\item the charges present give rise to a large potential difference between the hydrocarbon core and the outer water phase;
\item there are various dynamic phenomena, such as lipid diffusion in the bilayer plane.  
\end{itemize}
All these different bilayer characteristics, often involved in a complex interplay with proteins, are integral to many biological phenomena, such as transport, growth and signalling. 

     
%Fluidity is a central physical feature of membranes. 

%The fact that the lipid bilayers of almost all biological membranes are fluid under physiological conditions is almost certainly dictated by the requirement that most of the integral membrane proteins spanning the bilayer must undergo conformational transitions in order to function: membrane fluidity, in turn, implies {\it zero} shear restoring force.~\cite{lipow95a} <-- ... WHAT ABOUT THE LATERAL PRESSURE? SURELY IT DETERMINES DIFFERENT NON-ZERO RESTORING FORCES!!!


%\section[Transport Across Membranes]{Transport Across Membranes} There are three categories of membrane transport phenomena: passive, facilitated, and active. Here we will focus on the passive mechanism only, as this is the phenomenon that we have modeled as reported in Chpaters~\ref{ch:permSmall} and ~\ref{perm:Drugs}. The fundamental principle of passive permeation is described by Fick's first law of diffusion: a substance diffuses in the direction that eliminates its concentration gradient, at a rate proportional to the magnitude of this gradient. % \cite[p.727]{voet}


